Transfinite Induction/Principle 1

Theorem

Let $\On$ denote the class of all ordinals.

Let $A$ denote a class.

Suppose that:

For all elements $x$ of $\On$, if $x$ is a subset of $A$, then $x$ is an element of $A$.

Then $\On \subseteq A$.

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Aiming for a contradiction, suppose that $\neg \On \subseteq A$.

Then:

$\paren {\On \setminus A} \ne \O$

However, from the fact that $y$ is a strictly minimal element under $\in$ of $\On \setminus A$:

$\paren {\On \setminus A} \cap y = \O$

So by its subsethood of $\On$:

$\paren {\On \cap y} \setminus A = \paren {y \setminus A} = \O$

Therefore $y \subseteq A$.

However, by the hypothesis, $y$ must also be an element of $A$.

This contradicts the fact that $y$ is an element of $\On \setminus A$.

Therefore $\On \subseteq A$.

$\blacksquare$

This is one of the axioms of logic that was determined by Aristotle, and forms part of the backbone of classical (Aristotelian) logic.

This in turn invalidates this proof from an intuitionistic perspective.